In practice, it is simple to estimate the curvature at a
spectral peak using parabolic interpolation:

We can write

Note that the window ``amplitude-rate''
is always positive.
The ``chirp rate''
may be positive (increasing frequency) or
negative (downgoing chirps). For purposes of chirp-rate estimation,
there is no need to find the true spectral peak because the curvature
is the same for all
. However, curvature estimates are
generally more reliable near spectral peaks, where the signal-to-noise
ratio is typically maximum.
In practice, we can form an estimate of
from the known FFT
analysis window (typically ``close to Gaussian'').

Figure 10.25 shows the same chirplet in a time-frequency plot.
Figure 10.26 shows the spectrum of the example chirplet. Note
the parabolic fits to dB magnitude and unwrapped
phase. We see that phase modeling is most accurate where magnitude
is substantial. If the signal were not truncated in the time domain,
the parabolic fits would be perfect. Figure 10.27 shows the
spectrum of a Gaussian-windowed chirp in which frequency
decreases from 1 kHz to 500 Hz. Note how the curvature of the
phase at the peak has changed sign.